How to distort a generated node (not image) without using compositing?
Apologies.. I felt I had to make a tadpole with a longer tail to illustrate.
You can distort a procedural texture in exactly the same way as you would distort an image texture: by distorting the whole space containing it.

The straight object space, XY &gt; RG, and the procedural texture made in it
This is a straight tadpole: it was constructed as a set of functions and masks of ordinary Object space, because I thought it was easier to have 0 in the middle, but it could have been UV, or Generated space, shifted a bit, depending on how you want the space to behave.
To put, say, a wobble into the tail, you can manipulate the coordinates that the tadpole-making part of the tree uses, before you start making the tadpole, just as you might manipulate your given UV coordinates, before giving them to an Image texture to tell it where in the image to find a color.

The part of the tree which distorts the space, and f(X) as greyscale, which will be added to Y
To put a wobble into the Y of the procedural texture depending on X, you come up with some function of X that you can add to the Y coordinate, before handing the Y on to tadpole-making. Supposing it was just Y =&gt; Y + sin(X), that would put a sine-distortion in Y along the entire length of X, and you might have to mask off the head, to stop it being distorted, along with the tail.
This chunk of tree is just a more controlled version of that - it so happens it doesn't need a mask, because raising a negative number to a fractional power results in 0. The power also crunches the wave horizontally as the X goes positive. (Note our original Object X is multiplied by a negative before giving it to the power function, so positive X is then on the left)

The distorted space, and the resulting distorted tadpole
TL;DR
This is what happens to the Object space when the grey f(X) bands are added to its Y .. and the resulting distortion of the tadpole,when built using the X and Y's of the new space.
As you can see from the green value-field in the tree, you can keyframe the phase of the sine-function to animate the tail and move the head up and down, all in one.